The $Q_2$-free process in the hypercube

TL;DR

This paper introduces a process for generating $Q_2$-free subgraphs in hypercubes, providing probabilistic bounds on the number of edges and discussing conjectures and open questions in this new area.

Contribution

It defines the $Q_2$-free process in hypercubes and establishes a probabilistic lower bound on the resulting subgraph's edges, extending concepts from triangle-free processes.

Findings

With high probability, the resulting graph has at least $cd^{2/3} 2^d$ edges.

Heuristic arguments suggest a stronger conjecture about the process.

Discussion of challenges in rigorously proving the conjecture.

Abstract

The generation of a random triangle-saturated graph via the triangle-free process has been studied extensively. In this short note our aim is to introduce an analogous process in the hypercube. Specifically, we consider the $Q_2$-free process in $Q_d$ and the random subgraph of $Q_d$ it generates. Our main result is that with high probability the graph resulting from this process has at least $cd^{2/3} 2^d$ edges. We also discuss a heuristic argument based on the differential equations method which suggests a stronger conjecture, and discuss the issues with making this rigorous. We conclude with some open questions related to this process.